Commutators and implications of them not equalling 0

Question

I'm struggling to understand the commutator theory for quantum mechanics. I know there's the proof to do with $[P,Q]=0$ therefore there is a set of simultaneous eigenstates for $P$ and $Q$. However, if $[P,Q] neq 0$, does it also mean that then for all $|psirangle neq 0$, we have $[P,Q]|psirangle neq 0$? Or there is a way for it to equal 0?

Michael Seifert · Answer

As a counterexample: in general, the operators $hat{L}_x$ and $hat{L}_y$ do not commute, since $[hat{L}_x, hat{L}_y] = i hbar hat{L}_z$. However, for any state $|psirangle neq 0$ which is an eigenvector of $hat{L}_z$ with $L_z = 0$, we have $[L_x, L_y] |psirangle = 0$.

Answered by Michael Seifert on January 22, 2021

Commutators and implications of them not equalling 0

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